| L(s) = 1 | − 2i·2-s + 3.01i·3-s − 4·4-s + 6.03·6-s − 4.64i·7-s + 8i·8-s − 0.103·9-s − 12.0i·12-s − 9.29·14-s + 16·16-s + 0.206i·18-s + 14.0·21-s + 27.7i·23-s − 24.1·24-s + 26.8i·27-s + 18.5i·28-s + ⋯ |
| L(s) = 1 | − i·2-s + 1.00i·3-s − 4-s + 1.00·6-s − 0.664i·7-s + i·8-s − 0.0114·9-s − 1.00i·12-s − 0.664·14-s + 16-s + 0.0114i·18-s + 0.668·21-s + 1.20i·23-s − 1.00·24-s + 0.994i·27-s + 0.664i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.562626364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.562626364\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.01iT - 9T^{2} \) |
| 7 | \( 1 + 4.64iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 - 57.8T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 70.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 14.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 88.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 110.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 116iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 148. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 51.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.65597299235400057550320231039, −9.958547250193818274566627655871, −9.358174245665834552007140026817, −8.340832922103700799771423969637, −7.22226567277432975803495649259, −5.67488705417044301234940614888, −4.57268917892888542004264890500, −3.96513549072488105478306352153, −2.84670186037313727751863849172, −1.12907784161973989036442795744,
0.811991627840467214454278620642, 2.50438247123141558121141025297, 4.20143488098705652678007751225, 5.32053318534857455355752954820, 6.39229141303033894293825494099, 6.89526129396837304349492544684, 8.003054819792417505849718924440, 8.547364791352207747451490091428, 9.593042236329741097955851028247, 10.57716174238518090160657655189
